Consider the problem of constructing an experimental design, optimal for estimating parameters of a given statistical model with respect to a chosen criterion. To address this problem, the literature usually provides a single solution. Often, however, there exists a rich set of optimal designs, and the knowledge of this set can lead to substantially greater freedom to select an appropriate experiment. In this paper, we demonstrate that the set of all optimal approximate designs generally corresponds to a polytope. Particularly important elements of the polytope are its vertices, which we call vertex optimal designs. We prove that the vertex optimal designs possess unique properties, such as small supports, and outline strategies for how they can facilitate the construction of suitable experiments. Moreover, we show that for a variety of situations it is possible to construct the vertex optimal designs with the assistance of a computer, by employing error-free rational-arithmetic calculations. In such cases the vertex optimal designs are exact, often closely related to known combinatorial designs. Using this approach, we were able to determine the polytope of optimal designs for some of the most common multifactor regression models, thereby extending the choice of informative experiments for a large variety of applications.

翻译：考虑构建一个实验设计的问题，该设计对于给定统计模型的参数估计在选定准则下是最优的。为解决此问题，文献通常提供一个单一解。然而，通常存在一个丰富的最优设计集合，了解这一集合可以带来更大的自由度以选择合适的实验。本文证明，所有最优近似设计的集合通常对应一个多面体。该多面体中特别重要的元素是其顶点，我们称之为顶点最优设计。我们证明顶点最优设计具有独特的性质，例如较小的支撑集，并概述了它们如何促进构建合适实验的策略。此外，我们表明，对于多种情况，可以借助计算机通过采用无误差的有理数算术计算来构建顶点最优设计。在这种情况下，顶点最优设计是精确的，通常与已知的组合设计密切相关。利用这种方法，我们能够确定一些最常见的多因素回归模型的最优设计多面体，从而为大量应用扩展了信息性实验的选择范围。